Block Diagram Algebra in control system

Hello friends, in this blog article, we will learn Block diagram algebra in the control system. It will include block diagram reduction rules, some block diagram reduction examples and solutions.

We know that the input-output behavior of a linear system is given by its transfer function: G(s)=C(s)/R(s)

where R(s) = Laplace transform of the input variable and C(s) is Laplace transform of the output variable

A convenient graphical representation of such a linear system (transfer function) is called Block Diagram Algebra. 

A complex system is described by the interconnection of the different blocks for the individual components. Analysis of such a complicated system needs simplification of block diagrams by the use of block diagram algebra. Below table showing some of the rules for Block Diagram Reduction.

Block Diagram Reduction Rules

Block diagram reduction rules help you to minimize the block diagram thus solving the equations quickly. Below table represents block diagram reduction rules in the control system

Using the above rules you have to follow below simple steps to solve the block diagrams:

1. Combine all cascade blocks
2. Combine all parallel blocks
3. Eliminate all minor (interior) feedback loops
4. Shift summing points to left
5. Shift takeoff points to the right
6. Repeat steps 1 to 5 until the canonical form is obtained

Block diagram reduction examples

Now we will see some block diagram reduction examples. We will start with some simple examples and then will solve a few complex ones.

Example 1: In the below example, all the three blocks are in series (cascade). We just need to multiply them as G1(s)×G2(s)×G3(s).


Example 2: In this example, two blocks are in parallel but there is one summing point as well.


Example 3: Solve the below block diagram

Example 4: Simplify the block diagram shown in Figure below.


Step 1: Moved H2 before G2

Step 2: H1 and G2 are in parallel, thus added them as below

Step 3: (H1+G2) and G3 are in series, thus multiplied them

Step 4: Moved takeoff point 2 after G3(G2+H1)

Step 5: Minimizing parallel block with a feedback loop

Step 6: Finally, we will get the minimized equation as below

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